The Plancherel theory for Whittaker functions (see, for example, [1]}) implies that for \(x,y\in \mathbb {R}^N\) ,
\(\int _{\mathbb {R}^N} \mathrm {d}\lambda \, \psi _{-\lambda }(x) \psi _{\lambda }(y) m_N(\lambda )= \delta (x-y),\) 
